Density Definition and 1000 Threads

As I understand Darcy's law, it computes the flux of water through an area. The same value of flux can be the realized by different combinations of a density of water in the soil and the velocity of that water. So. as far as I can see, Darcy's law is ambiguous about which combination of...

Suppose that W(t) is just a Wiener process (i.e. a Gaussian in general). I want to know what the probability density for x, P(x), is. I started off by just assuming that I want to measure the expectation value of an observable f(x), so ##<f(x)>=\int_{W=0}^{W=t}{P(W)f(g(W))dW} \ \ ,\ \ x=g(W) ##...

With ##\rho=\sum_i p_i|\Psi_i\rangle\langle\Psi_i|##If the ##p_i=|\langle\Psi|\lambda_i\rangle|^2## are taken as joint probabilities given by quantum mechanics for the singlet state in EPRB then this cannot represent a statistical mix (classical) of those states because of Bell's theorem ?

1. If today vacuum and matter contribute 71 % and 29 % to the total energy density of the universe, at what redshift z were they contributing equally? 2. If today vacuum, matter, and radiation contribute 71 %, 29 %, and 0.01% to the total energy density of the universe, at what redshift z were...

Do we first find out as to what is the gravitational strength of a celestial object and then with that determine its density or do we first know what is its density and then conclude with that as to what is its gravity? Also whether the knowledge about gravitational strength or density comes...

In this explanation we need to involve the Dirac delta functions(maybe) but I clearly have a difficulty in understanding it can some one explain me the whole concept of constant or non constant volume charge density.

Hi. There has been a fair amount of research into electric generators and motors with superconductive coils. If traditional iron cores is used that obviously limits the power density because of the iron cores magnetic saturation point. But for coreless/ironless designs i don't understand what...

I don't understand the ratio [ 1 g / 6.02*10^23 amu] in the following conversion factor: [228.2 amu / 4.75*10^(-23) cm3] * [ 1 g / 6.02*10^23 amu] = 7.98 g/cm3 (This calculation is in a problem where you're finding the density of a unit cell in g/cm3 - though it has nothing to do with the...

Hi there, I've worked through most of this question but I'm stuck on the final part, showing that total bulk current ##I_B## is equal and opposite to total surface current ##I_S##. I calculated ##\vec H## the normal way I would if I was looking for ##\vec B## in an infinitely long cylindrical...

Summary:: Given statement: according to one model of the sun, the central mass density is 1.53x10^5kg.m^-3 and the mean opacity at the center is 0.217m^2kg^-1 Given statement: according to one model of the sun, the central mass density is 1.53x10^5kg.m^-3 and the mean opacity at the center is...

Hello! I was reading up on methods for determining the density of the universe and I came across this page: https://hypertextbook.com/facts/2000/ChristinaCheng.shtml I tried using equation stated, Ω=(2/3Λ)(c^2/H^2), with SI unit versions of both variables: Λ=1.1056 * 10^-52 m^-2 H=2.1927 *...

The answer is that the charge density would be -σ, I cannot for the life of me understand why would that be the case. Of course it makes sense but I can't convince myself that it would be the only possible answer. I have tried to apply Gauss law a few times, but it doesn't yield anything.

Does electric perm and magnetic perm give you enough info to work out OD? Thanks in advance

Electric field for the semi-circle $$E = - \frac {πKλ} {2R} $$ In this case E is equals to 10 N/C Electric field for the straighten wire $$E = 2Kλ * ( 1 - \frac {2y} {\sqrt{4y^2 + L^2}})$$ In this case E is equals to 8 N/C What I'm searching is R, λ, and the length of the wire, so I think...

I've been think about it for hours but I'm really out of clue here... The only things I could think of are obvious or useless... Any help would be greatly appreciated.

I read that density matrices are useful in Physics mainly to describe a) mixtures; we do not know the wave function of the system so ##\psi## is random and b) entangled systems. I'd like to focus on the later. Let us have a system ##S_1## entangled to the system ##S_2##. Thus we start from a...

UNITS m is meters kg is kiliograms K is degrees Kelvin s is seconds J is joules u is daltons = 1.66053906660(50)×10−27 kg https://en.wikipedia.org/wiki/Dalton_(unit) 1 pc = 3.085678 x 1016 m CONSTANTS MH = mass of hydrogen atom = 1.007825 u = 1.673532784796145 ×10−27 kg...

I'd like to show that, by minimizing this functional $$\Omega[\hat \rho] = \text{Tr} \hat \rho \left[ \hat H - \mu \hat N + \frac 1 {\beta} \log \hat \rho \right]$$ I get the well known expression $$\Omega[\hat \rho_0] = - \frac 1 {\beta} \log \text{Tr} e^{-\beta (\hat H - \mu \hat N )}$$ I'm...

2 scenarios: 1) Situation in the movie "2012", where volume of ocean water hasn't changed, but worldwide seismic activity has caused massive tidal waves. In one scene, a ship is sailing by Mt. Everest close to its peak. In such a scenario, I believe the air density outside the ship would the...

I'm not sure I understand why I need to use ##d##.. Maybe they want me to have the potential be zero at ##A##? In any case, I have found$$V(B)=\alpha k\int_0^L\frac{x}{\sqrt{b^2+\left(x-\frac{L}{2}\right)^2}}dx+C=\frac{\alpha...

Homework Statement:: The relative density of water is determined by the rate at which it expands (and contracts) with changes in temperature. At approximately what other temperature T does water have the same density as at 1 ∘C ? Relevant Equations:: Just looking at graphical and analyzing...

I attached a drawing of the problem for a better understanding and my attempted solutions. The first point is fairly simple but there's something that I can't figure out. dq=λdx=kxdx Q=∫ k x dx from 0 to L -> Q=k[x^2/2]0-L -> Q=(L^2/2)k -> k=2Q/L^2 This is what I came up with. I integrated...

Is the final answer given is correct? From where 96 (highlighted) came? Molar mass of $SO_2$ is $64.0588 g \times mol^{-1}$

Question 1; a) P=E/t E=5.796*10^7 J energy produced per day during the summer However, I am not certain how to calculate the time period, since although this concerns the energy produced per day, the sun does not shine for the entire duration of this 24 hour period. Also, I am unsure of the...

Answer : Using Pascal's law, this is my answer : ##\color{blue}{\boxed{\vec F_a = \vec F_c < \vec F_b}}##. Reasoning : Forces ##F_a## and ##F_c## are equal because the pressures required at the two cylinders for case (c) is the same as that required in (a). It doesn't matter how many of those...

I'm modelling the interiors of core-dominated (exo)planets. The EoS I use in my calculations are mostly either a Birch-Murnaghan formulation or a Mie-Grüneisen-Debye formulation. In either case, the ambient density ρ0 at ambient pressure and temperature are required for the implementation...

I am trying to understand an excerpt from an article describing the vibrations of a string (eg. guitar/piano) which reads as follows: This is basically the wave equation with Δm representing a small piece of mass from an interval of the string and two forces added to the right side. He...

There are some universe models where ##\Lambda < 0##. In this case, the energy density of the dark-energy becomes negative. At this point, does it make sense to talk about "negative dark energy density"? Or is it possible to think of this energy as curvature on space-time? Such that, ##\Lambda <...

I got the correct answer to this question with the following calculations, but I do need some correction in terms of what units I'm integrating across. ρ##\rho ## is density. mtot=n##m_{tot}=nM##, where n is the number of moles of a substance and M is the molar mass of the substance...

I have a question related to linearity of power spectral density calculation. Suppose I have a time series, divided into some epochs. If I compute PSD by Welch's method with a time window equal to the length of an epoch and without any overlap, I obtain this result: If I calculate the...

I am reading an article, which talks about graduated dark energy (gDE) model. In this model, it's assumed that the inertial mass density exhibits power-law dependence to its energy density $$\rho_{inert} = \gamma\rho_0(\frac{\rho}{\rho_0})^{\lambda}$$ Where ##\gamma## and ##\lambda## are real...

The energy density of an EM wave is given as (1/2) ϵ E^2 + (1/(2μ)) B^2. This is derived from the energy density of the electric and magnetic fields of capacitors and inductors, respectively. But why should the energy density of the fields of capacitors and inductors be the same as that of...

Hello all, Apologies in advance for the text-wall; this is a rather involved question. I am trying to compute the effective transmission coefficient for a medium of non-uniform refractive index. For simplicity I am assuming the slab has thickness ##d##, that ##n(0)=1##, and that ##n(d)=n##...

So this is a question from my lab report on capacitance. The aim of the experiment is to find out the relationship between surface charge density and radial distance from the centre of the plate capacitor. And in this experiment I have recorded 5 sets of data, namely r=0, V=4, r=1, V=3.5, r=2...

I have the following probability density function: $$f(x) =...

I have the solution for this problem but I did not understand the following statement: The mass of water the crown displaced is ##m = 740- 690= 50 g##. Therefore the volume of the crown is ## 50 cm^3## how can I conclude the volume of the crown from that displacement?

Hello, it's been a while since I've done any proper electrostatics, but I have a problem where I have a bunch of discrete point charges within some volume V bounded by a surface S. I am wondering if it is possible to replace the discrete charge density in my volume V by some continuous surface...

I couldn't solve the question. Can you help me?

I have the following probability density function (in Maple notation): f (x) = (1 / ((3/2) * Pi)) * (sin (x)) ** 2 with support [0; 3 * Pi] Now I want to transform x so that 0 -> (3/2) * Pi and 3 * Pi -> (15/2) * Pi and the new function is still a probability density function. How should I...

For getting the density of states formula for photons, we simply multiply the density of states for atoms by 2 (due to two spins of photons). I am getting the 2D density of states formula as :- g(p)dp = 2πApdp/h^2 I think this is the formula for normal particles, and so for photons I need to...

Given the support [a, b] of a probability density function. How can I change the formula for the probability density function with a support [u, v]? Example: Given the beta distribution with support [a=0,b=1]: $$\frac{x^{p-1} (1-x)^{q-1}}{Beta(p,q)}$$ Then the beta distribution with support...

My questions are as follows: 1. How do we find them and why do we need them? 2. What are the meanings of the mean and the median of a PDF? Are the formulae below correct? $$\int_{a}^{median} f(x) \mathrm{d}x = \int_{median}^{b} f(x) \mathrm{d}x$$ $$\int_{a}^{mean} f(x) \cdot x \mathrm{d}x =...

Why is electric current not a vector while electric current density is a vector? What's the intrinsic difference between the two through that surface integral?

After computind dirac 1D equation time dependant for a free particle particle I get 2 matrixs. From both,them I extract: 1) the probablity matrix P =ps1 * ps1 + psi2 *psi2 2) the current matrix J = np.conj(psi1)*psi2+np.conj(psi2)*psi1 I think that current is related to electricity, and...

Hello everyone, I have a math / physics question that has been with me for a while. I would be grateful if someone could help me. Given a density matrix, what is the minimum value a sum of some of its off-diagonal elements can assume (or the most negative value)? Remark: if one collect an...

What is column density in astronomy?

I think the right choice is c. I'll pass on my reasoning to you: We can think that if the formula of the potential is V(r)=\dfrac{kq}{r} If r tends to infinity, then V(r)=0. But the correct answer is d).

Cannot figure out where to start, apart from maybe something to do with specific heat capacity of water?

My guess is that the force per volume is: $$ \vec F_V = \rho \alpha x \hat x + \vec J \times \beta x \hat y$$ but I'm not sure where to go after that. I'm not given a value for either the charge density or the current density, so I can't simplify the relation much. Further, I'm not sure if my...

I have written a finite difference program to solve 1D time-independent Schrodinger equation. It seems to work correctly for harmonic oscillator, particle in a box, etc. But I can't figure out how to calculate the probability current density. It should be constant, but what is it? The program...